Aims. This paper is the first in a series investigating Lyman-alpha (hereafter Lyα) radiation transfer through hydrodynamical simulations of galaxy formation. Its aim is to assess the impact of interstellar medium (ISM) physics on Lyα radiation transfer and to quantify how galaxy orientation alters observational signatures with respect to the line of sight. Methods. We compare the results of Lyα radiation transfer calculations through the ISM of a couple of idealized galaxy simulations in a dark matter halo of ~1010 M⊙. In the first one, G1, this ISM is modeled using physics typical of large-scale cosmological hydrodynamics simulations of galaxy formation, where gas is prevented from radiatively cooling below 104 K. In the second one, G2, gas is allowed to radiate away more of its internal energy via metal lines and consequently fragments into dense star-forming clouds. Results. First, as expected, the small-scale structuration of the ISM plays a determinant role in shaping a galaxy’s Lyα properties. The artificially warm, hence smooth, ISM of G1 yields an escape fraction of ~50% at the Lyα line center, and produces symmetrical double-peak profiles. In contrast, in G2, most young stars are embedded in thick star-forming clouds, and the result is a ~10 times lower escape fraction. G2 also displays a stronger outflowing velocity field, which favors the escape of red-shifted photons, resulting in an asymmetric Lyα line. Second, the Lyα properties of G2 strongly depend on the inclination at which it is observed: From edge-on to face-on, the line goes from a double-peak profile with an equivalent width (EW) of ~−5 Å to a 15 times more luminous, red-shifted asymmetric line with EW ~ 90 Å. Conclusions. The remarkable discrepancy in the Lyα properties we derived for two ISM models raises a fundamental question. In effect, it demonstrates that Lyα radiation transfer calculations can only lead to realistic properties in simulations where galaxies are resolved into giant molecular clouds. Such a stringent requirement translates into severe constraints both in terms of ISM physics modeling and numerical resolution, putting these calculations beyond the reach of current large-scale cosmological simulations. Finally, we find inclination effects to be much stronger for Lyα photons than for continuum radiation. This could potentially introduce severe biases in the selection function of narrow-band Lyα emitter surveys and in their interpretation, and we predict that these surveys could indeed miss a significant fraction of the high-z galaxy population.